Integral Calculator
Evaluate definite integrals with the area shaded, and find antiderivatives of common forms
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Calculates definite integrals numerically with the area shaded on a graph, and finds antiderivatives of common forms with the working shown. Use it to check integration homework or to find the area under a curve.
How to use
- Type the function into "Function f", for example x^2 - 1, and check that "Read as" is correct.
- On the "Definite integral" tab, enter "Lower limit a" and "Upper limit b". Write inf or -inf for infinity.
- The value appears with the areas above and below the axis and, when an antiderivative is found, the F(b) − F(a) working.
- On the "Antiderivative" tab, read the result and "Steps: formulas and substitutions used".
How it works
- The definite integral is a numerical value from adaptive Gauss–Kronrod quadrature (the 7- and 15-point rules from QUADPACK), shown with its estimated error.
- For improper integrals, with an infinite limit or a point where the function is unbounded, convergence is judged numerically, so treat the result as an estimate.
- Antiderivatives cover common forms: polynomials, exponentials and logarithms, basic trigonometric forms, simple substitutions, basic integration by parts and partial fractions. Each result is differentiated back and checked before it is shown.
- Functions such as e^(-x^2) or sin(x)/x have no elementary antiderivative, but their definite integrals can still be calculated.
Questions and answers
- Is a definite integral the same as the area?
- No. A definite integral counts area below the axis as negative. For example, the integral of x^2 - 1 from 0 to 2 is 2/3, while the total area between the curve and the axis is 2.
- Why is there a + C in an antiderivative?
- The derivative of a constant is 0, so adding any constant to F(x) gives the same derivative. That is why an antiderivative is written F(x) + C.
- Does it show substitution and integration by parts?
- Yes. Simple substitutions such as 2x cos(x^2) and basic integration by parts such as x e^x or x^n ln x are shown with the method and substitution used.
- Why does it say the form is not supported?
- The function is outside the supported forms or has no elementary antiderivative. Press "Go to the definite integral" and enter limits to get a numerical value.